Over the past year, the majority of my research on this project was performed in four areas: (1) fitting a Hill model to dose-response data, (2) identifying situations in which Hill model parameters cannot be uniquely estimated, (3) generalizing the concept of relative potency, and (4) developing statistical methods for making inferences about generalized relative potency functions. These four areas of research are described in more detail below. Area 1: The Hill model is a popular nonlinear model often used to characterize dose-response data from toxicity studies. When the response data are binary, the Hill model corresponds to a 4-parameter logistic model. Compared to the usual 2-parameter logistic model, the Hill model allows the dose-response curve to have a lower limit that is greater than zero and an upper limit that is less than one. Thus, we can envision a scenario with three categories of subjects: those who will always respond regardless of dose (which leads to a lower limit above zero), those who will never respond regardless of dose (which leads to an upper limit below one), and those whose risk of response is a function of dose. We can postulate a missing data problem, where subjects were observed to respond or not, but we do not know which responders were "destined" to respond, nor do we know which non-responders were "unsusceptible" to response. We developed an EM algorithm to solve this missing data problem, which provides maximum likelihood estimates of the parameters in a Hill model for binary response. The EM algorithm is easy to program, covariates are simple to incorporate, and certain natural constraints are satisfied automatically. In ongoing research, we are investigating a similar approach for continuous dose-response data. We are again implementing an EM algorithm, but the analogy with responders who were "destined" to respond and non-responders who were "unsusceptible" to response no longer applies. In the continuous response situation, we envision two latent variables that are normally distributed and the sum of which equals the observed response. The mean of the first latent variable is a constant that corresponds to the lower limit of mean response. The mean of the second latent variable is the product of a dose-dependent probability and a constant that corresponds to the difference between the upper and lower limits of mean response. An EM algorithm can be used to estimate the parameters in this latent-variable model, which correspond to the parameters in a Hill model for continuous dose-response data. An article about the EM algorithm for binary dose-response data was published in the Journal of Agricultural, Biological, and Environmental Statistics. I also gave a talk about this research at the Joint Statistical Meetings of the American Statistical Association and the International Biometrics Society in Miami on March 22, 2011. Area 2: In related research, we are studying ways to determine the number of uniquely estimable parameters when a Hill model is fitted to binary data. It is well known that there are identifiability problems if all or none of the subjects respond. Similar problems arise if all subjects receiving a dose below a certain level do not respond and all subjects receiving a higher dose do respond. On the other hand, if there are several observed response rates and they tend to be ordered with respect to dose, then typically the Hill model parameters are uniquely estimable. For intermediate cases, however, the number of uniquely estimable parameters appears to be related to the number of distinct nonparametric estimates obtained under a monotonicity constraint on the dose-response curve (i.e., the number of level sets in a nonparametric isotonic regression analysis). Area 3: Relative potency plays an important role in toxicology. Estimates of relative potency are used to rank chemicals by their effects, to calculate equivalent doses of test chemicals compared to a standard, and to weight contributions of constituent chemicals when evaluating mixtures. Within a class of chemicals having "similar" dose-response curves, the relative potency of one chemical compared to another is the ratio of doses producing the same toxicity response, and this ratio is constant across all levels of response. If the dose-response curves are non-similar, however, relative potency need not be constant and typically varies according to where along the dose-response curves the dose ratio is calculated. In practice, relative potency is usually characterized by a constant dilution factor, even when non-similar dose-response curves indicate that constancy is inappropriate. Improperly regarding relative potency as constant may distort conclusions and potentially mislead investigators or policymakers. We developed a more general approach that allows relative potency to vary as a function of the dose of either chemical, the level of a specific response, or the percentage of the range of possible response levels. Distinct functions can be defined, each generalizing different but equivalent descriptions of constant relative potency. These relative potency functions are constructed from dose-response curves for test and reference chemicals, and they all provide fundamentally equivalent information if the chemicals have the same lower and upper limits of response. In fact, if two chemicals differ only with respect to their ED50s (i.e., their dose-response curves are similar), then all of the relative potency functions are constant and equal to the ratio of the ED50s. Otherwise, if the response limits differ, relative potency as a function of the response-range percentage is distinct from the other functions and embodies a modified definition of relative potency. Non-constant relative potency functions may cross the baseline value of 1.0, indicating that one chemical is more potent than another for some doses, responses, or response-range percentages and vice versa for others. If chemicals have non-similar dose-response curves, then inferences based on ratios of ED50s or based on models that force the other parameters to be identical can be misleading. Thus, we propose the use of relative potency functions, where the preferred function depends on the application (e.g., chemical ranking or dose conversion) and whether one views differences in response limits as intrinsic to the chemicals or as extrinsic, arising from idiosyncrasies of data sources. Relative potency functions offer a unified and principled description of relative potency for non-similar dose-response curves. We published an article about our generalized concept of relative potency in Regulatory Toxicology and Pharmacology. Also, I am scheduled to give an invited talk about this research at the Conference on Risk Assessment and Evaluation of Predictions to be held in the Washington DC area on October 12-14, 2011. Area 4: In ongoing research, we are working on formal statistical methods for analyzing relative potency functions. First, we will describe techniques for estimating parameters in the underlying dose-response models (e.g., Hill models), assessing model adequacy, quantifying variability of parameter estimates, constructing confidence intervals for model parameters, and testing hypotheses about model parameters. Then, based on specific models for the dose-response curves, we will develop procedures for making inferences about the resulting relative potency functions and any summaries obtained from these functions. These procedures will deal with function estimation, variance estimation, construction of pointwise confidence intervals and simultaneous confidence bands, and testing of hypotheses.